Advanced Calculus Single Variable

8.2 Operations On Power Series

It is desirable to be able to differentiate and multiply power series. The following theorem
says you can differentiate power series in the most natural way on the interval of convergence,
just as you would differentiate a polynomial. This theorem may seem obvious, but it is a
serious mistake to think this. You usually cannot differentiate an infinite series whose terms
are functions even if the functions are themselves polynomials. The following is special and
pertains to power series. It is another example of the interchange of two limits, in this case,
the limit involved in taking the derivative and the limit of the sequence of finite
sums.

When you formally differentiate a series term by term, the result is called the derived
series.

Theorem 8.2.1Let∑n=0∞an

(x− a)

nbe a Taylorseries having radius ofconvergence R > 0 and let

∞
f (x ) ≡ ∑ a (x − a)n (8.2)
n=0 n

(8.2)

for

|x − a|

< R. Then

∞∑ ∞∑
f ′(x) = ann(x− a)n−1 = ann(x − a)n−1 (8.3)
n=0 n=1

(8.3)

and this new differentiated power series, the derived series, has radius of convergence equal toR.

∕1!. Next use Theorem 8.2.1 again to take
the second derivative and obtain

∑∞
f′′(x) = 2a2 + ann (n− 1)(x− a)n−2
n=3

let x = a in this equation and obtain a2 = f′′

(a)

∕2 = f′′

(a)

∕2!. Continuing this way proves
the corollary. ■

This also shows the coefficients of a Taylor series are unique. That is, if

∑∞ k ∑∞ k
ak (x − a) = bk(x − a)
k=0 k=0

for all x in some open set containing a, then ak = bk for all k.

Example 8.2.3Find the sum∑k=1∞k2−k.

It may not be obvious what this sum equals but with the above theorem it is easy to find.
From the formula for the sum of a geometric series,

11−t

= ∑k=0∞tk if

|t|

< 1. Differentiate
both sides to obtain

∑∞
(1− t)− 2 = ktk−1
k=1

whenever

|t|

< 1. Let t = 1∕2. Then

∑∞
4 =-----1---- = k2−(k−1)
(1− (1∕2))2 k=1

and so if you multiply both sides by 2−1,

∞∑
2 = k2−k.
k=1

The above theorem shows that a power series is infinitely differentiable. Does it go the
other way? That is, if the function has infinitely many continuous derivatives, is it correctly
represented as a power series? The answer is no. See Problem 6 on Page 482 for an example.
In fact, this is an important example and distinction. The modern theory of partial
differential equations is built on just such functions which have many derivatives but no
power series.